Inequalities involving $\pi(x)$

Alzer, Horst; Kwong, Man Kam; Sándor, József

doi:10.4171/rsmup/98

Inequalities involving

π (x)

Alzer, Horst ; Kwong, Man Kam ; Sándor, József

Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251.

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Résumé

We present several inequalities involving the prime-counting function $π (x)$ . Here, we give two examples of our results. We show that

\frac{16}{9} π (x) π (y) \leq π^{2} (x + y)

is valid for all integers $x, y \geq 2$ . The constant factor $16 / 9$ is the best possible. The special case $x = y$ leads to

\frac{4}{3} \leq \frac{π (2 x)}{π (x)} (x = 2, 3, ...),

where the lower bound $4 / 3$ is sharp. This complements Landau’s well-known inequality

\frac{π (2 x)}{π (x)} \leq 2 (x = 2, 3, ...) .

Moreover, we prove that the inequality

{(2 \frac{π (x + y)}{x + y})}^{s} \leq {(\frac{π (x)}{x})}^{s} + {(\frac{π (y)}{y})}^{s} (0 < s \in ℝ)

holds for all integers $x, y \geq 2$ if and only if $s \leq s_{0} = 0.94745 ... .$ Here, $s_{0}$ is the only positive solution of

{(\frac{16}{7})}^{t} - {(\frac{6}{5})}^{t} = 1 .

Accepté le : 2020-07-20
Publié le : 2022-05-20

MR Zbl

DOI : 10.4171/rsmup/98

Classification : 11

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     author = {Alzer, Horst and Kwong, Man Kam and S\'andor, J\'ozsef},
     title = {Inequalities involving $\pi(x)$},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {237--251},
     volume = {147},
     year = {2022},
     doi = {10.4171/rsmup/98},
     mrnumber = {4450790},
     zbl = {1497.11014},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/rsmup/98/}
}

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Alzer, Horst; Kwong, Man Kam; Sándor, József. Inequalities involving $\pi(x)$. Rendiconti del Seminario Matematico della Università di Padova, Tome 147 (2022), pp. 237-251. doi : 10.4171/rsmup/98. http://archive.numdam.org/articles/10.4171/rsmup/98/

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